INTEGRABLE SYSTEMS RELATED TO BRAID GROUPS AND YANG-BAXTER EQUATION

1991 ◽  
pp. 135-149
Author(s):  
Toshitake KOHNO
Keyword(s):  
Integrable Systems ◽  
Braid Groups ◽  
Baxter Equation

scholarly journals Geometric Presentations of Braid Groups for Particles on a Graph

2021 ◽  
Author(s):  
Byung Hee An ◽  
Tomasz Maciazek
Keyword(s):  
Braid Group ◽  
Planar Graphs ◽  
Braid Groups ◽  
Quantum Statistics ◽  
Simple Cycle ◽  
Full Description ◽  
Baxter Equation ◽  
Connected Graphs ◽  
Star Graphs ◽  
Graph Theoretic

AbstractWe study geometric presentations of braid groups for particles that are constrained to move on a graph, i.e. a network consisting of nodes and edges. Our proposed set of generators consists of exchanges of pairs of particles on junctions of the graph and of certain circular moves where one particle travels around a simple cycle of the graph. We point out that so defined generators often do not satisfy the braiding relation known from 2D physics. We accomplish a full description of relations between the generators for star graphs where we derive certain quasi-braiding relations. We also describe how graph braid groups depend on the (graph-theoretic) connectivity of the graph. This is done in terms of quotients of graph braid groups where one-particle moves are put to identity. In particular, we show that for 3-connected planar graphs such a quotient reconstructs the well-known planar braid group. For 2-connected graphs this approach leads to generalisations of the Yang–Baxter equation. Our results are of particular relevance for the study of non-abelian anyons on networks showing new possibilities for non-abelian quantum statistics on graphs.

scholarly journals EXTRASPECIAL 2-GROUPS AND IMAGES OF BRAID GROUP REPRESENTATIONS

2006 ◽  
Vol 15 (04) ◽  
pp. 413-427 ◽  
Author(s):  
JENNIFER M. FRANKO ◽  
ERIC C. ROWELL ◽  
ZHENGHAN WANG
Keyword(s):  
Finite Groups ◽  
Braid Group ◽  
Braid Groups ◽  
Baxter Equation ◽  
Group Representations ◽  
Specific Solution ◽  
Link Invariants

We investigate a family of (reducible) representations of the braid groups [Formula: see text] corresponding to a specific solution to the Yang–Baxter equation. The images of [Formula: see text] under these representations are finite groups, and we identify them precisely as extensions of extra-special 2-groups. The decompositions of the representations into their irreducible constituents are determined, which allows us to relate them to the well-known Jones representations of [Formula: see text] factoring over Temperley–Lieb algebras and the corresponding link invariants.

PLANAR GENERALIZATIONS OF THE YANG-BAXTER EQUATION AND THEIR SKEINS

1992 ◽  
Vol 01 (02) ◽  
pp. 207-217 ◽  
Author(s):  
J. SCOTT CARTER ◽  
MASAHICO SAITO
Keyword(s):  
Coxeter Groups ◽  
Braid Groups ◽  
Baxter Equation

We solve the planar generalizations of the Yang-Baxter equation using Kauffman's bracket methods. These solutions are applied to get representations of certain Coxeter groups that are quotients of generalized braid groups.

scholarly journals A CATEGORICAL MODEL FOR THE VIRTUAL BRAID GROUP

2012 ◽  
Vol 21 (13) ◽  
pp. 1240008 ◽  
Author(s):  
LOUIS H. KAUFFMAN ◽  
SOFIA LAMBROPOULOU
Keyword(s):  
Braid Group ◽  
Hopf Algebras ◽  
Braid Groups ◽  
Point Of View ◽  
Baxter Equation ◽  
Quantum Invariants ◽  
Natural Group ◽  
Virtual Links ◽  
New Interpretation ◽  
Pure Virtual Braid Groups

This paper gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. The key to this approach is to take pure virtual braids as primary. The generators of the pure virtual braid group are abstract solutions to the algebraic Yang–Baxter equation. This point of view illuminates representations of the virtual braid groups and pure virtual braid groups via solutions to the algebraic Yang–Baxter equation. In this categorical framework, the virtual braid group is a natural group associated with the structure of algebraic braiding. We then point out how the category SC is related to categories associated with quantum algebras and Hopf algebras and with quantum invariants of virtual links.

scholarly journals BAXTER EQUATIONS AND DEFORMATION OF ABELIAN DIFFERENTIALS

2004 ◽  
Vol 19 (supp02) ◽  
pp. 396-417 ◽  
Author(s):  
F. A. SMIRNOV
Keyword(s):  
Integrable Systems ◽  
Classical Limit ◽  
Baxter Equation ◽  
Quantum Integrable Systems ◽  
Abelian Differentials ◽  
Bilinear Relation ◽  
Starting Point

In this paper the proofs are given of important properties of deformed Abelian differentials introduced earlier in connection with quantum integrable systems. The starting point of the construction is the Baxter equation. In particular, we prove the Riemann bilinear relation. The duality plays an important role in our consideration. The classical limit is considered in details.

scholarly journals Tetrahedron maps, Yang–Baxter maps, and partial linearisations

2021 ◽  
Author(s):  
Sergei Igonin ◽  
Vadim Kolesov ◽  
Sotiris Konstantinou-Rizos ◽  
Margarita Mikhailovna Preobrazhenskaia
Keyword(s):  
Integrable Systems ◽  
Soliton Solutions ◽  
Parametric Family ◽  
Nonlinear Dependence ◽  
Baxter Equation ◽  
Tetrahedron Equation ◽  
Matrix Groups ◽  
Linear Approximations ◽  
Kp Equations

Abstract We study tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation, and Yang–Baxter maps, which are set-theoretical solutions to the quantum Yang–Baxter equation. In particular, we clarify the structure of the nonlinear algebraic relations which define linear (parametric) tetrahedron maps (with nonlinear dependence on parameters), and we present several transformations which allow one to obtain new such maps from known ones. Furthermore, we prove that the differential of a (nonlinear) tetrahedron map on a manifold is a tetrahedron map as well. Similar results on the differentials of Yang–Baxter and entwining Yang–Baxter maps are also presented. Using the obtained general results, we construct new examples of (parametric) Yang–Baxter and tetrahedron maps. The considered examples include maps associated with integrable systems and matrix groups. In particular, we obtain a parametric family of new linear tetrahedron maps, which are linear approximations for the nonlinear tetrahedron map constructed by Dimakis and Müller-Hoissen [9] in a study of soliton solutions of vector Kadomtsev–Petviashvili (KP) equations. Also, we present invariants for this nonlinear tetrahedron map.

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